1. Field of the Invention
The present invention relates to information fusion, and more particularly to nonparametric information fusion for motion estimation.
2. Discussion of Related Art
Information fusion is important for many computer vision tasks. Information fusion is also important across modalities, for applications such as collision warning and avoidance, and speaker localization. Typically, a classical estimation framework such as the extended Kalman filter is employed to derive an estimate from multiple sensor data.
The problem of information fusion appears in many forms in computer vision. Tasks such as motion estimation, multimodal registration, tracking, and robot localization, typically use a synergy of estimates coming from multiple sources. However, typically the fusion algorithms assume a single source model and are not robust to outliers. If the data to be fused follow different underlying models, the conventional algorithms would produce poor estimates.
The quality of information fusion depends on the uncertainty of cross-correlation data. Let {circumflex over (χ)}1 and {circumflex over (χ)}2 be two estimates that are to be fused together to yield an optimal estimate {circumflex over (χ)}. The error covariances are defined byPij=E[(x−{circumflex over (x)}i)(x−{circumflex over (x)}j)]  (1)for i=1,2 and j=1,2. To simplify the notation denote P11≡P1 and P22≡P2.
Ignoring the cross-correlation, P12≡P21=0, the best linear unbiased estimator (BLUE), also called Simple Convex Combination, is expressed by:{circumflex over (x)}CC=PCC (P1−1{circumflex over (x)}1+P2−1{circumflex over (x)}2)  (2)PCC=(P1−1+=P2−1)−1  (3)
When the initial estimates are correlated P12≡P21≠0 and the noise correlation can be measured, the BLUE estimator ({circumflex over (χ)}BC,PRC) is derived according to Bar-Shalom and Campo using a Kalman formulation. The most general case of BLUE estimation also assumes prior knowledge of the covariance of χ.
A conservative approach to information fusion has been proposed by Julier and Uhlman in the form of the Covariance Intersection algorithm. The objective of the Covariance Intersection algorithm was to obtain a consistent estimator of the covariance matrix when two random variables are linearly combined and their cross-correlation is unknown. Consistency means that the estimated covariance is always an upper-bound, in the positive definite sense, of the true covariance, no matter what the cross-correlation level is. The intersection is characterized by the convex combination of the covariances{circumflex over (x)}CI=PCI(ωP1−1{circumflex over (x)}1+(1−ω)P2−1{circumflex over (x)}2)  (4)PCI=(ωP1−1+(1−ω)P2−1)−1  (5)where ω∈[0,1]. The parameter ω is chosen to optimize the trace or determinant of PCI.
Covariance Intersection has a very suggestive geometrical interpretation: if one plots the covariance ellipses P1, P2 and PBC (as given by the Bar-Shalom/Campo formulation) for all choices of P12, then PBC always lies within the intersection of P1 and P2. Thus, the strategy determines a PCI that encloses the intersection region and is consistent even for unknown P12. It has been shown in that the difference between PCI and the true covariance of χ is a semipositive matrix. More recently, Chong and Mori examined the performance of Covariance Intersection, while Chen, Arambel and Mehra analyzed the optimality of the algorithm.
Observe that the Covariance Intersection can be generalized to the fusion of n estimates as
                                          x            ^                    CI                =                              P            CI                    ⁢                                    ∑                              i                =                1                            n                        ⁢                                          ω                i                            ⁢                              P                i                                  -                  1                                            ⁢                                                x                  ^                                i                                                                        (        6        )                                          P          CI                =                              (                                          ∑                                  i                  =                  1                                n                            ⁢                                                ω                  i                                ⁢                                  P                  i                                      -                    1                                                                        )                                -            1                                              (        7        )            with
            ∑              i        =        1            n        ⁢          ω      i        =  1.In equations 6 and 7 the weights ωi are also chosen to minimize the trace or determinant of PCI.
Although important from theoretical viewpoint, Covariance Intersection has at least two weaknesses: it assumes a single source model and is not robust to outliers.
Therefore, a need exists for a system and method for information fusion that accommodates multiple source models and is robust to outliers.